The pro-EU Conservative lawmaker, Dominic Grieve, who has led efforts to hand parliament a greater say, said he was still looking forward to having discussions with the government to find a way out of a row that has questioned May's authority.
Leadsom told the Commons on Thursday she was "particularly sorry" that Shah "was forced to come and vote here while she was unwell" but then provoked anger from Labour MPs in the chamber because she sought to shift some of the blame to the opposition party.
British Prime Minister Teresa May called the adoption of the bill "an important step" in providing a "smooth" Brexit.
Leading Brexiteer Jacob Rees-Mogg told Sky News Mrs May would now attend a summit of European Union leaders next week "with full strength, with the ability to say the legislation to leave the European Union, under EU law and United Kingdom law, is now fully in place".
With both sides claiming victory, it is not always easy to establish who or if there was a clear victor; ultimately, however, Mrs May kept the core of the Brexit Bill the same.
In the end, MPs voted by 319 to 303 reject a rebel amendment to the EU (Withdrawal) Bill, which sets the legal framework for Brexit.
Theresa May has avoided a damaging Brexit defeat in the Commons after offering a last-minute concession to placate potential Tory rebels.
The British government is facing another knife-edge vote in Parliament on Wednesday June 20, 2018, over how much control lawmakers should have over the country's departure from the European Union.
Six Conservatives still voted against the government.
But while Mr Grieve withdrew support for his amendment, he still moved it to allow MPs to have their say.
Some of her opponents on Brexit may simply have made a decision to save their energy for later fights on issues such as future trading ties and customs arrangements with the bloc before Britain's scheduled departure in March next year.
Leadsom said a request that Shah be "nodded through" - a convention by which an ill MP can be deemed to have voted if they can be driven down to Westminster in a auto or ambulance - came too late when it was made at noon, four hours before the eventual vote.
On Wednesday evening, following the Commons vote, Leader of the Lords Baroness Evans of Bowes Park said the EU (Withdrawal) Bill had been "debated at length" and was very different as a result of amendments tabled by the Lords.
The government instead offered only an unamendable "neutral motion", which would have allowed MPs to take note of the situation.
Liberal Democrat Jo Swinson came into parliament to vote even though she is several days past the date she was due to give birth.
Grieve, who earlier said he woke up in the small hours worrying that his actions would bring down the PM, said after the vote: "We've managed to reach a compromise without breaking the Government - and I think some people don't realise we were getting quite close to that".